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In triangle PQR, A is the point of intersection of all the altitudes and B is the point of intersection of all the angle bisectors of the triangle. If ∠PBR = 1050, then what is the value of ∠PAR?
  • a)
    1500
  • b)
    1000
  • c)
    1050
  • d)
    1150
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In triangle PQR, A is the point of intersection of all the altitudes ...
∠PQR = 300
∠PAR = 1800 - 300 = 1500
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Most Upvoted Answer
In triangle PQR, A is the point of intersection of all the altitudes ...

Explanation:

- Given Information: ∠PBR = 1050
- Property of Angle Bisectors: In a triangle, the angle bisectors meet at a point called the incenter, which is equidistant from the sides of the triangle.
- Property of Altitudes: In a triangle, the altitudes meet at a point called the orthocenter, which is the point of concurrency of the altitudes.

Key Points:
- In triangle PQR, A is the orthocenter and B is the incenter.
- The angle bisector theorem states that the angle bisector divides the opposite side into segments proportional to the other two sides.
- Since B is the incenter, PB bisects angle QPR.

Solution:
- Since PB bisects angle QPR, we know that angle QPB = angle RPB = 1050/2 = 525.
- In triangle PBR, we have angle PBR + angle RPB + angle PRB = 180.
- Substituting the given angle PBR = 1050, we get angle PRB = 180 - 1050 - 525 = 450.
- Since A is the orthocenter, angle PAR = 90 - angle PRB = 90 - 450 = 150.
- Therefore, the value of ∠PAR is 1500.

Thus, option A is the correct answer.
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